Middle eastern / Islamic rejection of the Greco-Roman (which i'll call 'classical' from now on) tradition is a relatively new phenomenon.

It's important to remember that for hundreds of years until 1918 the Ottoman Empire ruled over nearly all Muslim populations around the Mediterranean and the Arabian peninsula. The rulers of the Seljuk Turks (whose vassals included the ancestors of the Ottoman founding dynasty) who seized Anatolia from the Byzantine Empire in the 11th century AD styled themselves as the Sultans of Rum (Arabic name for Rome). The Ottomans, when they came to power, were very conscious of the Roman tradition as well and exploited it to boost their legitimacy (e.g. moving capital to Constantinople in 15th century). Worth noting that the Ottomans' claim to the legacy of Rome played into their long-time rivalries with Venice and Austria (Austria re: its relationship to the Holy Roman Empire, and Venice insofar as the Venetians viewed themselves as hailing from more pure 'Latin' stock than the rest of Europe and thus the true successors of the Romans).

TL;DR: The Ottoman Empire did lay claim to the classical tradition and did so in the larger context of its general Occidental/European focus.

In thinking about the modern day rejection of the classical in the modern Arab world, I think there are two distinct but related driving factors: 1) the rise in the late 19th / early to mid 20th centuries of nationalism along ethnic lines, and 2) the post-1918 return of political Islam.

Nationalism you can look at, in the context of how it played out in the Middle East, as a reaction to the Ottoman cultural ethos. The Ottomans controlled as vast, multi-ethnic empire and regarded themselves as fundamentally European. They viewed themselves as the heirs to the Roman Empire and drew their authority from this presumption. By extension, when you get nationalism, when you get a desire for self-determination along ethnic lines, pan-Arabism for example, that implicitly involves a rejection of the classical tradition both as a rejection of Ottoman claims to authority in favor of Arab self-determination, and also, as a rejection of a foreign cultural tradition in favor of a uniquely Arab one.

Then you have political Islam. It works along the same lines as nationalism, except along religious rather than ethnic lines - e.g. rejection of foreign, implicitly Christian/heathen/sinful/unclean culture in favor of uniquely Islamic one etc. Goes even further than that though, you also get the Al Qaedas and Islamic States of the world who view themselves as the heirs to the tradition of the original caliphs who fought the Romans/Byzantine/Greeks. So not only do those ideologies reject the classical tradition and Europe in general, they view themselves as in a violent existential struggle with both.

There is a fantastic quote from one of my favorite books on Byzantine history by Mark Whittow, that I can only paraphrase because I don't remember exactly where it is.

Basically he said only the states already deeply committed to Christianity, i.e. Byzantium, remained so when the Arab conquests and their correspondingly victorious ideology spread quickly across the Mediterranean.

This is a useful quote because it illustrates that much of the foundational ideologies of "states," be they modern or ancient, and like nationalism for instance (which I know didn't technically exist at the time), is built upon "the last time they were great," and a belief that a reversion to the values and ideologies of their "last great time" will return them to greatness.

You see the echoes of this fundamentalism everywhere, both ancient and modern, from the Tea Party and the rhetoric of Trump, to the "caliphate" of the Islamic State, to the rhetoric of king Theodoric the Great, and emperors Aurelian and even Augustus. They all cloak themselves in their "renovatio."

So for the Middle East, they don't need to look further to their last great time than the era of the Rashidun caliphate. To the medieval europeans, they had to look further, to the Roman era.

But this is an argument on the basis of claimed (and thus invented) traditions. The values they seek to promote have to be seen as roughly in line with the existing ones now. But if you step aside from the question of claimed traditions, then in actuality, the non-political mechanics of a civilization, like urbanism, classical appreciation, governing systems, remained surprisingly continuous between the Roman to the Arab world.

It's just that the question of whether the Middle East WANTS to claim their heritage as Roman or not, is a political one. Given their preference for a closer Caliphate legacy and perceived conflicts with the west, which claims Roman heritage, they have clearly chosen not to lest they be seen now as political slaves to the west.

Because for someone to claim that the Middle East is "actually" Roman, when the West views itself as descended from the Romans, is to stake a nationalistic claim to the Middle East as belonging to the west. And that would be unacceptable to many in the Middle East.

History is never free from current politics.

I am a little unclear on the assertion in the latter half of the question. Can someone clarify it?

the fact that the Middle East has a much stronger connection to classical Greco-Roman tradition?

This a very strange claim. Most of the peoples of the Middle East are Arab/Turkish/Muslim/Persian/etc.... All of these are traditions with no origins in the Greco Roman world. Foreign conquerors who did adopt practices and ideas of the vanquished, but never made them part of their own like the Romans to a great extent did with the Greeks.

This question is like asking why Spain doesn't lay claim to the Umayyad tradition assuming that because they occupy the same region they have a stronger connection to them.

The truth is that a region and even the people in it are not a crucial factor in what makes a tradition continuous. The tradition of the Umayyads ended with Spain as the Roman tradition ended with the Arab and Turkish invasion. These new elites rarely looked at Rome for inspiration but Mecca instead.